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Almost Automorphic Solutions for the Heat Equation Involving the Discrete Fractional Laplacian in Continuous and Discrete Time
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DOI：10.1002/mma.70678.png)
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Abstract
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In this work, we study the almost automorphic property of solutions to the one-dimensional fractional heat equation involving the discrete fractional Laplacian with in the Lebesgue space , considering both continuous and discrete time cases. The lattice system is formulated as an initial value problem, whose solutions are expressed through a subordination formula involving the continuous and discrete L & eacute;vy functions, as well as the semi-discrete heat kernel defined in terms of the modified Bessel functions. We establish sufficient conditions to guarantee the existence and uniqueness of almost automorphic solutions under suitable Lipschitz-type assumptions, relying on fixed point theorems. To achieve this, we prove invariance under convolution and the superposition principle for the almost automorphic property.
Keywords:
discrete almost automorphic
discrete fractional Laplacian
full discrete heat equation
semidiscrete heat equation
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